1^0+1^0+3^0=3;1^1+1^1+3^1=5;1^2+1^2+3^2=11;1^3+1^3+3^3=29;1^4+1^4+3^4=83; 113 is the smallest (non-trivial) prime for which the sums of the first 5 powers of the digits are all prime. [Rupinski]

The largest of the fifteen prime numbers p that require a prime number of digits when written in any prime base less than or equal to p. It is 1110001 in base 2, 11012 in base 3, 423 in base 5, 221 in base 7, and requires two digits in bases 11, 13, 17,... 113.
[Hartley]

The three circularly permuted primes, i.e., 113, 131 and 311, can be concatenated in 3! different ways. After inserting a prime digit as a separator between these concatenated primes, the following four 11-digit primes can be obtained: 11371317311, 31151315113, 13153115113 and 11373117131. Note that the first two numbers are palindromic primes. [Necula]

The smallest prime of the form (p^2 + q^2 + r^2)/3, where p, q, r are 3 consecutive primes. [Pfoertner]

It is possible to form seven sums of primes with the prime value of 113 using the first 22 primes: 2+19+31+61 = 3+37+73 = 5+41+67 = 7+47+59 = 11+23+79 = 13+29+71 = 17+43+53 = 113. [Gallardo]

(Joke) Theorem: Any number is prime. Proof: by the alphabet code, the phrase "any number" is 113. [Gallardo]

113 is the smallest prime such that the sum of the squares
of the digits (11) is prime and the product of the digits
(3) is also prime. Note that the concatenation of the two
prime values (11 and 3) gives the prime number 113. [Capelle]

Longitude 113 degrees West passes through Dolphin Island in Great Salt Lake, Utah. Think of it as a "prime meridian" for the world's saltiest sailors.

The greatest known prime number p for which there is no
prime number between p and p + sqrt(p). [Capelle]

The smallest prime that yields another if the sum of digits
of its consecutive digits is sandwiched between each of the
corresponding consecutive digits (i.e., 113 becomes 12143). [Silva]

The recently published AINUR (the Atlas of Images of Nuclear Rings) in the journal Monthly Notices of the Royal Astronomical Society covers 113 nuclear rings in 107 [also prime] different galaxies. [Post]

The smallest prime number which is the sum of three fourth powers of primes. It is 2^4 + 2^4 + 3^4. [Hartley]

(n) is defined as the number of primes less than or equal
to n. The prime number theorem states that (n)*ln(n)/n
approaches 1 as n tends to infinity. The maximum value of
this expression, 1.2550587..., occurs at n = 113. [Keith]

113 divided into (5*71) ≈ . Note that 113, 5, and 71, are all prime. [Cirillo]

An FBI investigation determined that 113 emails on Clinton’s servers contained information that was classified at the time.

The star Arcturus actually shines 113 times more brightly than our
Sun, if you were able to compare them side by side. [Becker]